Long-term discount rates do not vary across firms

Transcription

1 Long-term discount rates do not vary across firms Matti Keloharju Juhani T. Linnainmaa Peter Nyberg April 2018 Abstract Long-term expected returns appear to vary little, if at all, in the cross section of stocks. We devise a bootstrapping procedure that injects small amounts of variation into expected returns and show that even negligible differences in expected returns, if they existed, would be easy to detect. Markers of such differences, however, are absent from actual stock returns. Our estimates are consistent with production-based asset pricing models such as Berk, Green, and Naik (1999) and Zhang (2005) in which firms risks change over time. Our results imply that stock market anomalies have only a limited effect on firm valuations. Keloharju is with Aalto University School of Business, CEPR, and IFN. Linnainmaa is with the University of Southern California and NBER. Nyberg is with the Aalto University School of Business. We thank Jonathan Berk, John Cochrane, Mathias Kronlund, Anders Löflund, Owen Lamont, and conference and seminar participants at the LA Finance Day, Florida International University, University of Illinois at Chicago, University of Illinois at Urbana-Champaign, University of Miami, and University of Michigan for insightful comments and Jonathan Berk, Timothy Johnson, and Lu Zhang for providing the code for the models simulated in Section 2. Financial support from the Academy of Finland and Nasdaq Nordic Foundation is gratefully acknowledged.

2 1 Introduction Time-varying risks lie at the heart of production-based asset pricing models. As firms make new investments and old sources of cash flows turn obsolete, the riskiness of the firms asset bases changes (Berk, Green, and Naik 1999; Gomes, Kogan, and Zhang 2003). And when firms face systematic productivity shocks and costly reversibility of investment, high-risk firms can turn into low-risk firms, and vice versa (Zhang 2005; Cooper 2006). We show that these models share one common feature: cross-sectional differences in expected returns vanish over time. This result does not depend on the specific mechanism driving time-varying risks. In this paper we measure the extent to which the data align with this unifying feature of productionbased asset pricing models. We first show that the data strongly support the idea of expected returns converging over time. Even if a stock s expected return today lies far above or below the average, we cannot reject the null that its expected return after five years equals the average. Second, we show that the convergence in expected returns is commensurate with that in risks, regardless of whether we measure them using betas or firm characteristics. Over time, today s high-risk firms become less risky and low-risk firms become riskier. In Panel A of Figure 1 we use U.S. stock data and assign stocks into deciles each month from 1963 through 2016 based on estimates of stocks expected returns. In this computation, which we detail below, our proxy for firms expected returns uses a combination of 34 return predictors. We report the average returns for the top and bottom deciles over the next ten years. The average returns we report are forward rates; that is, the month-t estimate is the average return in month t after portfolio formation, not the average return from today to month t. Panels B and C illustrate the extent to which the data conform to two models, Berk, Green, and Naik (1999) and Gomes, Kogan, and Zhang (2003). 1

3 Figure 1: Average monthly returns on stocks sorted by expected returns. Panel A assigns U.S. stocks into deciles based on a combination of 34 return predictors that use accounting information (see text for details) and plots the average (non-cumulative) monthly returns for the top and bottom deciles over the next ten years. Panels B and C simulate data from Berk et al. s (1999) and Gomes et al. s (2003) models using cross-sectionally demeaned returns. The number of stocks and time periods in the simulations match the average number of stocks and number of months in the U.S. data. We simulate data from these models using the original papers parameters and rank firms into deciles each month based on their expected returns. In both the simulations and actual data, differences in average returns are initially large but collapse to zero as we extend the maturity. Expected returns behave differently in the two models. In Berk, Green, and Naik (1999), the cross-sectional distribution of expected returns is almost symmetric and the differences decay slowly; in Gomes, Kogan, and Zhang (2003), the distribution is left-skewed and the differences decay faster. We illustrate our key finding first with a characteristics-free bootstrap procedure. We take the cross section of monthly stock returns from 1963 through 2016, preserve both the covariance structure and distribution of returns, but set all expected returns to zero. We then inject small cross-sectional differences into expected returns and measure our ability to detect these differences. If differences in expected returns persist, past stock returns positively predict the cross section of stock returns. To see why, note that a stock s return, r it, here consists of a constant term, µ i, plus noise, ε it : r it = µ i + ε it. (1) 2

4 A stock s average return therefore measures its expected return: r itn = 1 n n r i,t j = µ i j=1 }{{} expected return n ε i,t j. + 1 n j=1 }{{} average noise term (2) A cross-sectional regression of stock returns against past average returns is a regression of returns against noisy estimates of expected returns. 1 We show that returns would be highly predictable even if the persistent differences in expected returns were negligible. Suppose, for example, that the crosssectional standard deviation of the expected monthly stock returns is just 0.4%. This is a small number, as it would imply that differences in expected returns would account for just R 2 = 0.06% of the cross-sectional variance of realized stock returns. 2 Our bootstrap procedure shows that, among allbut-microcaps, the prior five-year return would nevertheless predict the cross section of returns with a t-value of In actual data, past returns do not positively predict returns outside the prior one-year period (momentum). In the five-year regression, for example, the t-value is As in Figure 1, we complement the bootstrap methodology by sorting stocks into portfolios by different combinations of return predictors. Many variables predict the cross section of stock returns, 3 but most are short-lived. Although some predict returns a few years out, none of them identify longterm differences in expected returns. We show that this property extends to combinations of predictors as well. When we sort stocks into portfolios by a combination of 46 predictors, the average return 1 Conrad and Kaul (1998) suggest that this mechanism that the cross section of realized returns measure differences in expected returns may account for some of the momentum in stock returns: The repeated purchase of winners from the proceeds of the sale of losers will, on average, be tantamount to the purchase of high-mean securities from the sale of low-mean securities. Consequently, as long as there is some cross-sectional dispersion in the mean returns of the universe of securities, a momentum strategy will be profitable. See, also, Berk, Green, and Naik (1999, p. 1584). 2 The average cross-sectional variance of U.S. stock returns between 1963 and 2016 is If the cross-sectional standard deviation of expected returns is 0.4%, this variation in expected returns would account for (0.004)2 = 0.06% of the cross-sectional variation in stock returns; or, conversely, 99.94% of the cross-sectional variation in monthly stock returns would be unrelated to persistent differences in expected returns. 3 See, for example, Hou, Xue, and Zhang (2015), McLean and Pontiff (2016), and Harvey, Liu, and Zhu (2016). 3

5 difference between the top and bottom deciles is 0.74% in the year following portfolio formation (t-value = 4.48); but, in the second year, the return difference is just 0.25% with a t-value of Even when we select ex-post the most persistent predictors which turn out to be those that use accounting rather than price, volume, and return information the combination of these predictors loses its predictive power after approximately three years. The long-run return estimates are precise enough to bound the amount of cross-sectional variation in expected returns. Consider, for example, the CAPM alphas. In the year following portfolio formation, the 95% confidence interval for the CAPM alpha for the difference between the top and bottom deciles runs from 77 basis points to 129 basis points. In year eight, this confidence interval runs from 14 basis points to 31 basis points. We can thus reliably identify significant differences in short-term expected returns but, as Figure 1 shows, these differences evaporate quickly. In production-based asset pricing models, expected returns converge because risks mean revert. We therefore also examine how firms change over time. Without taking a stance on whether betas or characteristics better measure risk, we show that both converge in expectation towards the mean. However, whereas differences in average returns collapse to zero in five years, those in betas and characteristics do not. We show that a decomposition of firm characteristics into permanent and transitory components resolves the seeming discrepancy in the behaviors of average returns and characteristics. Persistent cross-sectional differences in characteristics are not associated with any differences in average returns. It is the transitory differences in characteristics that command premiums and discounts, and these differences vanish in approximately five years. These results tie back to those in Cohen and Polk (1996), Asness, Porter, and Stevens (2000), and Novy-Marx (2013), who find that industry-level (and therefore persistent) differences in characteristics such as value are not associated with differences in 4

6 average returns. We generalize this result by showing that persistent differences in characteristics, no matter the source, do not correlate with differences in expected returns. Existing literature documents sizable short-term differences in expected returns. 4 We contribute to the literature by showing that there is a wide gap between short-term and long-term returns and little evidence of persistent differences in expected returns between firms. Because any differences in expected returns are short-lived, they carry little weight in the average discount rate and in firm valuation. 5 These results are consistent with financial markets valuing the equity of different firms using approximately the same discount rate. Although the data are consistent with the predictions of production-based asset pricing models, we cannot rule out other theories either. Consumption-based asset pricing models are the flip side of production-based asset pricing models (Cochrane 1991). If firms covariances with consumption growth change over time and their risks are stationary (that is, they mean revert in the cross section), firms expected returns will, on average, converge towards the mean. The data also do not shut out behavioral theories. If differences in expected returns reflect mispricing, and arbitrageurs profit by trading against mispricing, cross-sectional differences in expected returns can be expected to vanish over time. What the data reject are models in which differences in expected returns persist to a significant degree. Assuming that all anomalies reflect mispricing gives insight into mispricing s potential economic significance. As pointed out by van Binsbergen and Opp (2017), more persistent anomalies are likely to generate greater distortions in the economy. An anomaly that generates an expected return of 10% per annum and lasts for a year causes prices to be wrong by 10%; an otherwise similar anomaly lasting only for one month causes prices to be wrong by less than one percent. If most anomalies are short-lived as 4 For example, Martin and Wagner (2016) estimate that there is considerably more variation in expected returns... than has previously been acknowledged. 5 If a firm s discount rate is r over the first year and r thereafter, its k-year discount rate is [ (1 + r)(1 + r ) k 1] 1/k 1. Differences in short-run discount rates carry over to long-run discount rates, but increasing the horizon k dilutes their role. 5

7 our estimates suggest the stock market can be inefficient in returns but close to efficient in prices. Its perceived efficiency depends on the investment horizon. An arbitrageur could reap great rewards by trading anomalies but, at the same time, the market would be close to efficient to a buy-and-hold investor. 2 Expected returns in production-based asset pricing models Many production-based asset pricing theories seek to explain cross-sectional return patterns associated with, for example, size and book-to-market. These models feature time-varying risks. In this section we describe and simulate data from four models. We show that and explain why cross-sectional firm-level differences in expected returns in these models tend to vanish. 2.1 Models and mechanisms Model 1: Berk, Green, and Naik (1999): assets in place versus growth options. Firms encounter new projects each period and they accept those with positive NPVs. Projects differ in their amount of systematic risk, and old projects turn obsolete at random. Because new projects are drawn from the same distribution, firms are asymptotically identical. In other words, high-risk firms will, on average, encounter projects that lower their risk, and low-risk firms will tend to encounter projects that increase their risk. Firm valuation in the model buttresses this point. A firm s value is the sum of the value of the assets in place and the value of future growth options. Of these two terms, the value of the future growth options is the same across firms, because all firms expect to encounter the same projects. Model 2: Gomes, Kogan, and Zhang (2003): general equilibrium with growth options. Gomes, Kogan, and Zhang (2003) take the investment mechanism of Berk, Green, and Naik (BGN, 1999) to general equilibrium. Whereas BGN assume the process describing the pricing kernel, Gomes, 6

8 Kogan, and Zhang (2003) model the household sector and let the markets clear. Returns in this model are completely described by a conditional CAPM; size and book-to-market predict returns because they correlate with the true conditional market betas. Also similar to BGN, new projects are distributed randomly across all firms with equal probabilities. Therefore, all firms derive the same value from future growth options and expected returns perfectly converge over time. Model 3: Zhang (2005) and Lin and Zhang (2013): production shocks and costly reversibility of investment. Production in this model requires capital, and firm-level productivity is subject to aggregate and idiosyncratic shocks. Firms have to pay to install new capital and to adjust it. The pricing kernel is parametrized directly to have countercyclical price of risk. Productivity shocks alter firms riskiness; in bad times, when the price of risk is high, low-productivity firms find it costly to shed unproductive capital. Because productivity shocks mean revert, firms expected returns converge over time. Model 4: Hackbarth and Johnson (2015) and Gu, Hackbarth, and Johnson (2017): operating leverage and real options. Similar to Zhang (2005), production in this model requires capital, and productivity is subject to aggregate and idiosyncratic shocks. Firms face both quasi-fixed and variable costs for both upward and downward adjustments to capital. Firms invest more only when productivity attains a high enough level and disinvest if it falls to a sufficiently low level. Risks associated with operating leverage and real options move in opposite directions. As a firm s market-to-book increases, its assets in place become less risky but its expansion and contraction options become more risky. The relationship between market-to-book and expected return is therefore nonmonotonic. 7

9 2.2 Changes in expected returns We simulate 1,000 months of return data from the models described above using the parameters used in the original studies; when a study considers multiple sets of parameters, we use those of the baseline specification. Studies use different methods to choose the parameters. They are typically fixed based on prior literature, directly estimated, or calibrated to match some features of the data, such as the levels and volatilities of equity premium and interest rate. We discard the first 400 months of data to ensure that the simulations stabilize. We assign stocks into deciles based on expected returns at the end of month t, and compute average returns for stocks in these deciles over the next 15 years. We then average the estimates over all starting months t. We repeat each simulation 1,000 times to reduce simulation-specific noise; this is different from the graphs in the introduction s Figure 1 in which we plot the data from single runs. Figure 2 shows average, cross-sectionally demeaned monthly returns from the four models discussed above. The models differ in the amount of dispersion in expected returns. In Berk, Green, and Naik (1999), for example, the difference in expected monthly returns between the top and bottom deciles is just over 40 basis points; in Zhang (2005), this difference is over 140 basis points. In some models, such as Berk, Green, and Naik (1999), the cross-sectional distribution of expected returns is nearly symmetric; in others, such as Gomes, Kogan, and Zhang (2003), it is considerably left-skewed. The common element of these models, however, is the convergence in expected returns. Although the models differ in the speed of convergence they model different economic mechanisms, and they are parametrized differently expected returns converge in all of them. Expected returns in these models compensate for one or multiple sources of risk. Figure A1 in the appendix shows that, as expected returns converge towards the mean, so do market betas. 8

10 Figure 2: Average monthly returns on stocks sorted by expected returns. We simulate 1,000 months of return data from the four models described in Section 2. We run these simulations using the same parameters as those used in the original studies. We discard the first 400 months and then begin ranking stocks into deciles based on expected returns. We report the average cross-sectionally demeaned monthly returns for these deciles over the next 15 years after portfolio sorts. The pattern in Figure 2 is not specific to the models we consider. If risks change but are stationary, then they must mean-revert. Moreover, unless a model builds in permanent cross-sectional differences in, for example, production technology, each firm s risk (and its expected return) must be expected to converge towards the common mean. We are not aware of any study on production-based asset pricing models that has found the need to build in persistent differences across firms. 9

11 3 Data We use the daily and monthly CRSP return data from January 1963 through December 2016 on stocks listed on the NYSE, Amex, and NASDAQ. We exclude securities other than ordinary common shares. We also exclude financials, which are identified as firms with SIC codes between 6000 and We use CRSP delisting returns; if a delisting return is missing and the delisting is performance-related, we impute a return of 30% for NYSE and Amex stocks (Shumway 1997) and 55% for Nasdaq stocks (Shumway and Warther 1999). We use balance sheet and income statement information from the annual and quarterly Compustat files to construct various return predictors that have been proposed in the literature. We describe these predictors in Section Detecting differences in long-term discount rates: A bootstrap approach 4.1 Cross-sectional regressions In this section we estimate regressions that predict the cross section of stock returns with each stock s average past return. Each month t we estimate a cross-sectional regression r it = a t + b t r i,t k1,t k 2 + e it, (3) where r i,t k1,t k 2 is stock i s average return from month t k 2 to t k 1. If stock returns contain persistent differences in expected returns, as in equation (1), the slope estimate from these regressions 10

12 is proportional to cross-sectional variance of these differences, ˆb ˆσ 2 µ. For example, if expected returns are constant and return innovations IID, then ˆbt = covcs (r it, r i,t k1,t k 2 ) var cs ( r i,t k1,t k 2 ) = covcs 1 (µ i + ε i,t, µ i + var cs 1 (µ i + k 2 k 1 +1 t k1 k 2 k 1 +1 t k1 t =t k 2 ε i,t ) t =t k 2 ε i,t ) = ˆσ 2 µ ˆσ 2 µ + 1 k 2 k 1 +1 ˆσ2 ε 0. (4) We estimate these regressions using actual and bootstrapped stock return data. We construct the bootstrapped data to preserve both the distributions of returns and their covariance structure. 4.2 Methodology We generate each draw of simulated data in five steps: 1. We draw each stock s expected return from a normal distribution with a mean of zero and a standard deviation σ µ. We denote stock i s expected return by µ i. 2. We cross-sectionally demean month-t returns and divide them by their standard deviation to generate a vector of residuals with unit variance. We denote this N t 1 vector of residuals by ɛ t, where N t is the number of stocks in month t. 3. We estimate the N t N t covariance matrix of stock returns, Σ t, using monthly data from month t 30 to month t + 30, that is, five years plus one month. We estimate pairwise covariances; that is, we do not require all stocks to have non-missing returns for the entire estimation period. Because the number of stocks exceeds the length of the time series, Σ t is singular. 4. We replace the covariance matrix Σ t with its nearest positive definite matrix using the algorithm of Higham (2002). We denote this positive definite matrix by S t. 5. We generate month-t stock-specific shocks ε t by randomizing the elements of ɛ t and by post- 11

13 multiplying them by the Cholesky factor of S t. Stock i s return in month t is then µ i + ε it. These simulated returns have appealing properties. First, the resulting data matrix has the same dimensions as the actual return data, that is, it has the same number of months and the same number of stocks each month. Second, the factor structure of returns is the same as that in the actual data and, given the rolling estimates of covariances, this factor structure changes over time as it does in the data. Third, because we extract the shocks from actual stock returns, the distribution of stock returns each month resembles the actual distribution of returns. We draw 100 random samples of returns using this procedure. Using each simulated dataset, we estimate the cross-sectional regressions and record the average regression slopes and their t-values. We then compare these estimates to those from the actual data, and examine how the estimates change as we increase the amount of variation in expected returns, σ µ. 4.3 Estimates Table 1 reports results from cross-sectional regressions that we estimate using both actual and simulated data. Panel A uses data on all stocks; Panel B restricts the sample to all-but-microcaps. Allbut-microcaps are stocks with market capitalizations above the 20th percentile of the NYSE distribution. The estimates in the first column of Table 1 use actual data; the remaining columns simulate data with values of σ µ ranging from 0% to 1.4%. The average cross-sectional standard deviation of realized monthly stock returns is 16.9%. Therefore, when σ µ = 1%, cross-sectional variation in expected returns explains (0.01) 2 /(0.169) 2 = 0.35% of the cross-sectional variation in realized returns. We estimate the cross-sectional regressions in equation (3) using various past return windows. The first row predicts the cross section of month t returns using month t 1 returns; the last row uses prior twenty-year returns skipping a month. Each regression contains stocks that have non-missing returns 12

14 Table 1: Fama-MacBeth regressions with actual and bootstrapped data This table reports average coefficients and t-values from regressions to predict the cross section of monthly stock returns. The explanatory variable is the stock s average return over the window specified in the first column. Column Actual data uses returns on common stocks listed on NYSE, Amex, and NASDAQ from 1963 through The sample excludes financials, which are identified as firms with SIC codes between 6000 and Columns Bootstrapped data use data that are first randomized to set the variation in expected returns to zero. These data are generated in five steps. First, month-t returns are demeaned and divided by their standard deviation to generate a vector of residuals, ɛ t, with unit variance. Second, the covariance matrix of individual stock returns, Σ t, is estimated using data from 30 months prior to 30 months after month t. Third, this covariance matrix is transformed to the nearest positive definite matrix S t using the algorithm of Higham (2002). Fourth, month-t returns are generated by randomizing the elements of ɛ t and by post-multiplying this vector by the Cholesky factor of S t. Fifth, µ i is added to stock i s return each month, where µ i s are drawn from a distribution with a mean of zero and a standard deviation of σ µ. The simulations vary the σ µ from 0 to 1.4%. We repeat the bootstrapping procedure 100 times for each value of σ µ ; this table reports the average coefficients and t-values across these simulations. Panel A uses data on all stocks; Panel B uses all-butmicrocaps. All-but-microcaps are stocks with market capitalizations above the 20th percentile in the NYSE distribution. Panel A: All stocks Historical Bootstrapped data return Actual Cross-sectional variation in expected returns, σ µ horizon data 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% Fama-MacBeth coefficient estimates [ 1, 1] [ 12, 2] [ 60, 13] [ 120, 61] [ 120, 2] [ 240, 121] [ 240, 2] t-values [ 1, 1] [ 12, 2] [ 60, 13] [ 120, 61] [ 120, 2] [ 240, 121] [ 240, 2]

15 Panel B: All-but-microcaps Historical Bootstrapped data return Actual Cross-sectional variation in expected returns, σ µ horizon data 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% Fama-MacBeth coefficient estimates [ 1, 1] [ 12, 2] [ 60, 13] [ 120, 61] [ 120, 2] [ 240, 121] [ 240, 2] t-values [ 1, 1] [ 12, 2] [ 60, 13] [ 120, 61] [ 120, 2] [ 240, 121] [ 240, 2] in month t and for at least half of the past return period. The regressions on the last row, for example, include stocks that have at least ten years of non-missing returns. The estimates that use actual return data show the effects of short-term reversals (Jegadeesh 1990), momentum (Jegadeesh and Titman 1993), and long-term reversals (De Bondt and Thaler 1985). Shortterm reversals show up as negative slope coefficients in cross-sectional regressions of month t returns against prior month returns. The estimates in the full and all-but-microcaps samples are 0.05 (t-value = 12.12) and 0.02 (t-value = 3.51). Momentum registers as positive coefficients in regressions against prior one-year returns skipping a month. The estimates in the full and all-but-microcaps samples are 0.04 and 0.10, and these estimates are associated with t-values of 2.54 and Long-term reversals appear as negative slope coefficients when we lag average returns by more than a year. For example, in 14

16 regressions that predict returns using prior five-year returns skipping a year, the slope coefficients are 0.17 and 0.11, and the t-values are 6.43 and Data in which expected returns are constant over time, vary across firms, and stock-specific innovations are serially and cross-serially uncorrelated cannot produce negative regression slopes. The inevitability of positive regression coefficients under these assumptions is best illustrated by considering the Lo and MacKinlay (1990) decomposition of trading profits. 6 Let us consider a strategy that places a weight of w i,t = 1 N ( r i,t k 1,t k 2 r m,t k1,t k 2 ) (5) on stock i, where r i,t k1,t k 2 is stock i s average return from month t k 2 to t k 1, and r m,t k1,t k 2 is the return on the equal-weighted market index. Following Lo and MacKinlay (1990) and Lewellen ( N ) (2002), the expected trading profit in month t, E(π t ) = E i=1 w i,tr i,t, then decomposes into E(π t ) = N 1 N 2 tr(σ) 1 N 2 [ 1 Σ1 tr(σ) ] + σ 2 µ, (6) where tr(σ) is the the trace of the covariance matrix (which is the sum of the autocovariances), 1 Σ1 tr(σ) is the sum of the covariance matrix s off-diagonal elements, and σ 2 µ is the cross-sectional variance of unconditional expected returns. The last term is always positive; persistent cross-sectional variation in mean returns increases the profitability of trading strategies in which stock weights increase in realized returns. Profits can therefore be negative only if stock returns are negatively autocorrelated (the first term of equation (6) is negative) or positively cross-serially correlated (the second term of equation (6) after the minus sign is positive). That is, a negative slope coefficient emerges only if a high return on an asset predicts a low return on that asset or high returns on other assets. 6 Lo and MacKinlay (1990) decompose profits to strategies that trade long-term reversals. Lewellen (2002) considers profits to strategies that trade the momentum in individual stocks or portfolios of stocks. 15

17 The estimated regression slope in equation (4), which assumes IID innovations, is nonnegative; the IID assumption sets the first two terms of the decomposition in equation (6) to zero. Therefore, in Table 1 s bootstrapped data, the slope coefficients start at zero when σ µ = 0 and become increasingly positive as we increase the amount of cross-sectional variation in expected returns. When σ µ = 0.4%, average past returns over various horizons begin to be identified as statistically significant predictors of returns in the full sample. For the [ 240, 2] window in Panel A, for example, the average slope coefficient is 0.17 and the t-value associated with this estimate is In the all-but-microcaps, the estimates turn statistical significant already when σ µ = 0.2%. The simulations in Table 1 show that if there was even a small amount of persistent variation in expected returns, our regressions would have the power to detect it. A volatility parameter of σ µ = 0.4%, for example, corresponds to a world in which a cross-sectional regression of realized month-t returns r it against expected returns µ i has an R 2 of 0.06%. Moreover, if expected returns remained constant in the cross section, past returns should become more informative about returns as we increase the length of the past-return window; a wider window yields more precise estimates of expected returns. In the simulations the t-values indeed increase in the length of the estimation window. In the data, however, the slope coefficients are and remain negative after the one-year momentum period. These comparisons suggest that the amount of persistent cross-sectional variation in expected returns must be negligible; whatever the variation might be, it is completely overshadowed by long-term reversals in individual stock returns. The term persistent can also be interpreted loosely here. If stocks expected returns change over time, but slowly, then a regression against five-year average returns would still typically return a positive coefficient. In the data, the cutoff is one year. It is instructive to consider momentum to understand the amount of predictive power we would expect to find at longer lags if cross-sectional differences in expected returns persisted. In Panel B s 16

18 all-but-microcaps sample, the slope coefficient on momentum is 0.10 with a t-value of In the bootstrapped data, we approximately match this coefficient when σ µ = 0.1%. Here, the slope coefficient is 0.06 with a t-value of If we were to attribute momentum to persistent cross-sectional variation in mean returns, long-horizon past returns would be tremendously powerful predictors of the cross section of stock returns. For example, in regressions against the prior ten-year returns skipping a month, the t-value would be Portfolio sorts 5.1 Return predictors We complement the bootstrap analysis by sorting stocks into portfolios using 46 predictors of stock returns. We use different combinations of these predictors to form portfolios and then examine the persistence in these portfolios average returns. The predictors we include are among those analyzed in McLean and Pontiff (2016). We also take those few additional predictors from Linnainmaa and Roberts (2017) that were published after McLean and Pontiff (2016) created their list. Table 2 reports monthly average returns and CAPM and three-factor model alphas for HML-style factors based on each of the 46 return predictors. We sort stocks into six portfolios by market capitalization and the predictor and then compute value-weighted returns on these portfolios. The breakpoint for size is the 50th NYSE percentile and those for the predictor are the 30th and 70th NYSE percentiles. A factor s return is the average return on the two high portfolios minus the average return on the two low portfolios. 17

19 Table 2: Average returns, CAPM and three-factor model alphas, and persistence of 46 return predictors This table reports average returns and CAPM and three-factor model alphas for 46 return predictors. Each return predictor is used to construct an HML-style factor by sorting stocks into six portfolios by size and predictor. These sorts are independent and use NYSE breakpoints; the size breakpoint is the median and the predictor breakpoints are the 30th and 70th percentiles. A predictor s return is the average return on the two value-weighted high portfolios minus the average return on the two value-weighted low portfolios. The high and low portfolios are determined so that high corresponds to those stocks that the initial study identified as earning higher returns. Accounting-based predictors are rebalanced annually at the end of each June; Return-based predictors are rebalanced monthly. Persistence is the last holding period after which the average return or the CAPM or the threefactor model alpha has a p-value above 0.05 for three consecutive months. A value of 50, for example, indicates that the factor s average return or alpha remains statistically significantly different from zero for 50 months after portfolio formation. Average CAPM FF3 Persist- Start return alpha alpha ence, Predictor year r t ˆα t ˆα t months Accounting-based predictors Earnings to price Enterprise multiple Gross profitability Inventory growth Piotroski s F score Abnormal investment Accruals Accruals and book-to-market Net operating assets Net working capital changes O-score Profit margin Asset growth Sales to price One-year share issuance Five-year share issuance Sustainable growth Total external financing Z-score Industry-adjusted CAPX growth Sales-minus-inventory growth Investment to capital Investment growth rate Investment to assets

20 QMJ: Profitability Distress Book to market Operating profitability Organization capital Cashflow to equity Return on assets Return on equity Asset turnover Debt issuance Return-based predictors 52-week high Amihud s illiquidity Beta Idiosyncratic volatility Industry momentum Long-term reversals Maximum daily return Momentum Intermediate momentum Seasonality Short-term reversals High-volume return premium We divide the predictors into accounting- and return-based predictors. A predictor is accountingbased if it uses any information from either the balance sheet or income statement; the return-based predictors use only return, price, or volume information. We recompute and rebalance the accountingbased factors annually at the end of June and the return-based factors monthly. The 46 return predictors that we examine explain differences in average returns over our 1963 through 2016 sample period. Each factor earns a statistically significant average return, CAPM alpha, or threefactor model alpha. The last column reports estimates of how long a factor retains its statistically significant average return or alpha. We define persistence as the number of months that can be skipped after portfolio formation without losing statistical significance for three consecutive months. Inventory growth, for example, has an estimated persistence of seven months. That is, this factor s returns 19

21 are statistically significant in months one through seven after portfolio formation, but statistically insignificant afterwards. 7 We later use these persistence estimates to group together the ex-post most persistent predictors. 5.2 Average long-term returns In Figure 3 and Table 3, we form value-weighted portfolios using different combinations of return predictors and then examine differences in average returns at various horizons. In these computations, the portfolios are formed at date t and then held unchanged for up to ten years. In addition to using all 46 returns predictors, we form portfolios based on three subsets of predictors: (a) the 34 accountingbased predictors, (b) the 12 return-based predictors, and (c) the 21 ex-post most persistent predictors. These are the predictors that retain their statistical significance for at least over a year according to Table 2 s last column. We form the portfolios by computing the average standardized predictor for each stock. We first convert each predictor into a z-score by subtracting the cross-sectional average and by then dividing by the cross-sectional standard deviation (Asness, Frazzini, and Pedersen 2013). Similar to the factors in Table 2, we sign each predictor so that high values correspond to high average returns based on the original study. A stock s signal is the average of its non-missing z-scores. Figure 3 suggests that there are little, if any, persistent differences in average returns. In this figure we take the difference between the top and bottom deciles and report the average value-weighted monthly returns (Panel A) and CAPM alphas (Panel B) for months 1 to 120 following portfolio formation. These are non-cumulative returns. That is, the return at horizon k is the average return that an investor would have earned in month k after portfolio formation ( forward rate ). Table 3 reports average returns and 7 Predictors can regain statistical significance later by chance or, in the case of seasonality, by nature (Heston and Sadka 2008). If a predictor is statistically insignificant for three consecutive months, Table 2 classifies it as having lost its predictive power. 20

22 Panel A: Average monthly returns Panel B: Monthly CAPM alphas Figure 3: Average monthly returns and CAPM alphas on long-short strategies in months after portfolio formation. We form decile portfolios at the end of each month by sorting stocks into portfolios by different combinations of the 46 return predictors listed in Table 2. The left hand side of each panel uses the 34 predictors that use income statement or balance sheet information; the right hand side of each panel uses the 12 predictors that use price, return, or volume information. We convert each predictor into a z-score by subtracting the cross-sectional average from the predictor and by dividing the difference by the cross-sectional standard deviation. Each stock s predictor is the average of its non-missing z-scores. We construct value-weighted portfolios each month and hold these portfolios for up to ten years. Panel A plots the average return difference between the top and bottom deciles, while Panel B plots the monthly CAPM alphas for the return difference between the top and bottom deciles. The average return in month t is its month t return ( forward rate ), not its average return from today to month t. The shaded areas indicate 95% confidence intervals. 21

23 Table 3: Measuring differences in average long-term returns We form decile portfolios at the end of each month by sorting stocks into portfolios by different combinations of the 46 return predictors listed in Table 2. Row All uses all predictors; Accounting-based uses the 34 predictors that use some income statement or balance sheet information; Return-based uses the 12 predictors that use price, return, or volume information; and Ex-post long-lived uses the 21 predictors in Table 2 that, when used in isolation, persist for at least a year and a month. We convert each predictor into a z-score by subtracting the cross-sectional average from the predictor and by dividing the difference by the cross-sectional standard deviation. Each stock s predictor is the average of its non-missing z-scores. We construct value-weighted portfolios each month and hold these portfolios for up to eight years. Panel A reports average monthly returns and t-values for value-weighted high-minus-low portfolios for different horizons following portfolio formation; Panel B reports monthly CAPM alphas and t-values. The holding periods range from one month after portfolio formation to year ten after portfolio formation. We adjust standard errors for overlapping returns using the Jegadeesh and Titman (1993) method. Panel A: Average returns Return Horizon predictor 1 month Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Average monthly long-short returns All Accounting-based Price-based Long-lived t-values All Accounting-based Price-based Long-lived t-values for the month following portfolio formation, as well as the average monthly returns and CAPM alphas in years 1, 2,..., 10 following portfolio formation. We adjust standard errors for overlapping observations using the Jegadeesh and Titman (1993) procedure. Panel A of Table 3 shows that the average return differences between the top and bottom quintiles rapidly vanish when we estimate expected returns using all 46 predictors. 8 One month after portfolio 8 Figure 3 and Table 3 use value-weighted portfolios. Table A1 in the Appendix reports the estimates for equal-weighted portfolios. 22

24 Panel B: CAPM alphas Return Horizon predictor 1 month Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 CAPM alphas All Accounting-based Price-based Long-lived t-values All Accounting-based Price-based Long-lived formation, the average return for the long-short portfolio is 123 basis points (t-value = 6.62); in the year following portfolio formation, the return difference is 74 basis points (t-value = 4.48); and, in year two, it is just 25 basis points (t-value = 1.55). After year two, the differences in average returns are economically small and statistically insignificant. Taken together, this full set of predictors is not informative about long-term differences in average returns. Figure 3 shows that accounting-based predictors are significantly more informative about differences in long-term returns than return-based predictors. The accounting-based predictors generate a return difference of 40 basis points per month in year three (t-value = 2.88) while the return-based predictors are statistically significant only in the first year. The difference is consistent with the persistence estimates in Table 2. Moreover, this difference between the two groups of predictors is also consistent with prior research. Some return-based predictors, such as short-term reversals, are known to last only for about one month (Jegadeesh 1990; Goyal and Wahal 2015). Others, such as momentum, turn into reversals at longer lags (Jegadeesh and Titman 1993). Although some other predictors, such as idiosyncratic volatility, could contain information about long-term returns, they do not. 23

25 The last row of Table 3 shows that even when we select the ex-post most persistent predictors those that, on their own, are statistically significantly different from zero for more than a year following portfolio formation we cannot reliably detect differences in average returns between the top and bottom quintiles after year three. This test purposefully stacks the playing field in favor of finding differences in average returns; after all, we use the same data to both identify the most promising predictors and to test their performance. Nevertheless, this set of predictors displays only the same amount of persistence as the full set of accounting-based predictors. Panel B of Figure 3 and Panel B of Table 3 report monthly CAPM alphas for the long-short strategies from Panel A. Statistically significant differences in CAPM alphas persist longer than those in average returns they converge to zero seven years after portfolio formation. Market adjustment helps both by increasing point estimates and, by the virtue of removing market-wide variation, lowering standard errors. When we use all 46 return predictors, the CAPM alpha in the first year after portfolio formation is 103 basis points with a standard error of 13 basis points; in Panel A, by contrast, the average return in year one is 74 basis points with a standard error of 17 basis points. The fact that CAPM alphas are higher than average returns is consistent with Table 2 s result that most anomalies are stronger on a market risk-adjusted basis. Average returns (Panel A) and market-adjusted returns (Panel B) are both important, but for different purposes. Market-adjusted returns are relevant from an investing viewpoint, while average returns are the ones that matter for capital budgeting. An investor could manage his portfolio to keep it market neutral, thereby aiming to earn the market-adjusted returns reported in Panel B. 9 Firms, however, do not discount cashflows back at market-adjusted rates. Instead, Panel A s average returns represent the appropriate estimates of the differences in firms (forward) discount rates. 9 A statistically significant CAPM alpha indicates that an investor who currently holds the market could earn a higher Sharpe ratio by additionally taking a position in the left-hand side strategy (Huberman and Kandel 1987). 24